Optional node set for the dominant-chamber fold and Borel–Weil–Bott

[pbelmans]

Requested by HomogeneousTools/PartialFlagVarieties.jl#21: reflecting in only a subset S of the simple roots, so a weight can be folded into the dominant chamber of a Levi subgroup rather than of the whole group.

API

An optional trailing nodes argument, defaulting to nothing, which means all of them, on

• conjugate_dominant_weight
• conjugate_dominant_weight_with_elem
• conjugate_dominant_weight_with_length
• borel_weil_bott
• is_singular

Everything routes through the two existing kernels _fold_dominant! and _fold_dominant_with_word!. They still scan 1:R and now test membership in nodes, rather than iterating over the node list. That matters for two reasons:

• every index into the coordinate vector and the Cartan matrix still provably lies in range, so a caller-supplied node can never read past the end under @inbounds. Indexing a caller-supplied node list did exactly that: under the default --check-bounds=auto an out-of-range node silently returned a garbage weight, and only --check-bounds=yes — which Pkg.test() forces, so the suite is blind to it — threw. It is now an ArgumentError, checked once at the public boundary;
• nodes may be any container supporting in: tuple, vector, range or Set. Duplicates are harmless.

nothing marks the full fold, and being a singleton type its membership test const-folds away, so there is one code path and the hot one is unchanged.

For borel_weil_bott the singularity test is restricted to nodes as well, which makes it the relative statement along G/B → G/P_I with nodes the nodes unmarked in I. Note ρ = ρ_G stays the right shift even in the restricted case, since ρ_G - ρ_S pairs to zero with every coroot in S and is therefore W_S-invariant; that is recorded in the docstring.

No regression on the default path

_fold_dominant! is the kernel every cohomology, dimension and Euler-characteristic call funnels through downstream, so the default path was measured, not assumed. 2000 random E₈ weights, minimum of a @benchmark:

	min	median
main	0.3145 ms	0.3492 ms
this branch, 4 runs	0.3141 – 0.3284 ms	0.3491 – 0.3525 ms

Straddles the main figure, and results are identical. The nodes === nothing test const-folds after inlining, as the comment on the kernel claims.

Tests

Full suite green, doctests green. 1000 new assertions in three testsets, over types A, B, C, D, G₂, F₄, E₆:

• the restricted fold agrees with the same fold carried out inside the sub-root-system via sub_dynkin_type_with_ordering — the ground truth for the whole change;
• the result is S-dominant, the fold is idempotent, the word from _with_elem lies in S, has length equal to _with_length, and replaying it reproduces the folded weight;
• coordinates at nodes not adjacent to S are untouched;
• restricted borel_weil_bott matches the absolute statement inside the sub-diagram, degree and weight, including agreement on when it vanishes;
• any node container giving the same answer as a tuple, and an out-of-range node throwing from all five entry points;
• is_singular(λ + ρ, S) against the ground truth of pairing λ + ρ with every positive root of the subsystem, and against borel_weil_bott(λ, S) === nothing, plus the implication that singular for S forces singular for G;
• weights regular for G but singular for S and vice versa, and the degenerate empty node set.

is_singular restricted to nodes is exactly the vanishing criterion of the relative borel_weil_bott, and the tests pin those two against each other.

Follow-up, not in this PR

• Once this is on main, PartialFlagVarieties.jl can delete its local _fold_dominant_in and call borel_weil_bott(λ, S) directly, and its Semisimple compat bound moves to the release carrying this.

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